Yes they do, it’s longbows and it’s horribly anachronistic.
No, I’m suggesting the exact same Sharpshooter as the Americans get. They’re roughly equivalent to Longbowmen in nearly every aspect. The only difference is that Longbowmen perform much better against skirmishers/archers and require the Yeomen card to stay on par with them. Sharpshooters only pull ahead with the Paper Cartridge upgrade and the fact that they wouldn’t need to send a card to get to that level (but they do need to spend more resources on the Counter Infantry Rifling upgrade to keep on par with longbows). On maps with Seminole, Longbows could actually be slightly better due to the Seminole Bowyers upgrade.